Worm gear design: lead angle, self-locking, ratio and efficiency

How a worm drive works and its ratio

A worm looks like a screw and the worm wheel like a throated helical gear, on shafts crossed at 90°. Each worm revolution advances the wheel by as many teeth as the worm has starts, so a single stage reaches ratios of 5–100 — ideal for lifts, rotary tables, valve actuators and winches.

i = z₂ / z₁　　d₁ = q · m　　d₂ = m · z₂　　a = (d₁ + d₂) / 2 = m · (q + z₂) / 2

z₁ is the number of worm starts, z₂ the wheel teeth, m the module (worm axial module = wheel transverse module) and q the diameter quotient. With m2, q10, a single-start worm and a 40-tooth wheel: i = 40, worm pitch diameter 20 mm, wheel 80 mm, centre distance a = 50 mm. A two-start worm halves the ratio to 20.

TIPWorm and wheel must have the same hand (right with right) and matching module, pressure angle and lead angle. The worm wheel generator derives the helix angle from the worm's pitch diameter and starts.

Calculating the lead angle

Axial pitch px = π · m　　Lead pz = z₁ · px　　tan γ = pz / (π · d₁) = z₁ / q

The lead angle γ — between the thread and a plane normal to the worm axis — governs efficiency and self-locking. More starts and a smaller q give a larger lead angle. Lead angles for the common q values (ISO / JIS recommend 8, 10, 12.5, 16):

Small q means a larger lead angle and higher efficiency but a slender, flexible worm; large q gives a stiff worm that self-locks more easily but is less efficient. A slender worm under heavy load bends and shifts the mesh, which limits how small q can go.

Self-locking: tan γ < μ

When the wheel tries to drive the worm, friction on the thread flank may be enough to prevent sliding — the drive self-locks and the load holds without a brake. The condition is a lead angle below the equivalent friction angle:

ρ' = atan(μ / cos αn)　　Self-locking: γ ≤ ρ' (roughly tan γ < μ)

• At rest, friction is higher (steel on bronze ≈ 0.1–0.15, ρ' ≈ 6°–9°), so single-start worms with γ below about 5° usually lock statically.
• Running, with an oil film, μ can drop to 0.03–0.05 and ρ' to about 2°–3°. Vibration or shock can then let the wheel creep backwards (no dynamic self-locking).
• A self-locking worm always has forward efficiency below 50%: at γ = ρ', η = tan ρ' / tan 2ρ' < 0.5. Self-locking costs efficiency and heat.

TIPLubrication, temperature, vibration and wear all affect self-locking. For hoists, lifts and anything safety-related, add a brake or mechanical lock — never rely on self-locking alone.

Estimating efficiency

Forward η = tan γ / tan(γ + ρ')　　Back-driving η' = tan(γ − ρ') / tan γ (≤ 0 means self-locking)

μ = 0.03: hardened steel worm, phosphor-bronze wheel, good lubrication and higher sliding speed; 0.05: normal lubrication; 0.1: start-up or poor lubrication; 0.1–0.2: greased 3D-printed plastic worms. These are mesh efficiencies — bearings and oil churning take another 2–5%.

Sliding speed vs = π · d₁ · n₁ / (60000 · cos γ)　(m/s)

For d₁ = 20 mm at 1500 rpm, vs ≈ 1.58 m/s. Higher sliding speed builds a better oil film and lowers μ, but generates more heat. All lost power becomes heat: 100 W in at 65% means 35 W the housing must shed.

Forces and design steps

T₂ = T₁ · i · η　　Ft₂ = Fa₁ = 2T₂ / d₂　　Ft₁ = Fa₂ = 2T₁ / d₁

The wheel's tangential force equals the worm's axial force — often large — so the worm shaft needs thrust-capable bearings (angular-contact, tapered roller or thrust bearings), located in both directions for reversing drives.

1. Choose ratio i and starts z₁: one start for self-locking, two or more for efficiency; wheels usually have 28 teeth or more.
2. Choose module m and quotient q; compute γ = atan(z₁/q).
3. Compute centre distance a = m(q + z₂)/2 and check the envelope.
4. Estimate η, output torque T₂ and worm thrust; decide whether self-locking is required.
5. Worm thread length b₁ ≥ (11 + 0.06·z₂)·m; wheel face width about 0.7–0.75 × worm tip diameter.
6. Check heat and lubrication; continuous-duty gearboxes need a thermal rating.

Example: a NEMA 17 at 0.4 N·m drives an m2, q10 single-start worm and 40-tooth wheel. γ = 5.71°; at μ = 0.05, η ≈ 65%, so T₂ = 0.4 × 40 × 0.65 ≈ 10.4 N·m. Worm thrust Fa₁ = 2 × 10.4 / 0.08 ≈ 260 N; thread length ≥ (11 + 2.4) × 2 ≈ 27 mm; centre distance 50 mm.

Materials, lubrication and 3D printing

• The industrial pairing is a hardened, ground steel worm with a phosphor-bronze wheel — hard against soft, so wear goes to the replaceable wheel.
• Worm meshes slide, so lubrication matters far more than for spur gears: EP gear oil in an oil bath, or grease for small mechanisms.
• For 3D printing use m ≥ 1 and q = 8–10, a POM-like, nylon or PETG wheel and grease. PLA softens under frictional heat.
• Printed upright, the underside thread flank is a steep overhang — use ≤ 0.15 mm layers and strong cooling, or SLS / MJF nylon for precision.
• Align the worm axis with the middle of the wheel face. Centre-distance errors cause backlash or binding, so allow 0.1–0.2 mm of adjustment.